Hi there,
In my studies I come up with this nonconvex optimization problem
argmin |Ax|_2+lamda*|x|_1 subject to x'x=1
where cost function is nonsmooth but convex and the constrant in nonconvex.
I tries subgradient projection method for convex constraints but the global solution is not my desired solution.
My question is that I should solve this problem hurestically or there is a reliable method for this nonconvex optimization problem?

2 Answers
2

You can have a look of these papers：
1. Jonathan H. Manton, Optimization algorithms exploiting unitary constraints.
2. Zaiwen Zai and Wotao Yin, A feasible method for optimization with orthogonality constraints.

The objective function is now differentiable, so without further ado you can invoke the Gradient-Projection method, which under reasonable assumptions can be guaranteed to converge.

This formulation makes it easy to use Alternating-Projection approaches.

Of course, several other numerical ideas also apply. For example, to get a good solution, you could start with $\gamma$ very large so that the $\ell_1$ constraint essentially disappears; then solve the problem exactly, and then gradually tighten $\gamma$.

This trick seems interesting but I have no explicit data term in misfit functional and I am solving system of homogenous equations and data term is in the A matrix. So I have problem in using Alternating-Projection approaches with no b in the misfit functional.
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user26030Aug 29 '12 at 20:00